{$\cal W$}-Gauge Structures and their Anomalies:An Algebraic Approach

Starting from flat two-dimensional gauge potentials we propose the notion of ${\cal W}$-gauge structure in terms of a nilpotent BRS differential algebra. The decomposition of the underlying Lie algebra with respect to an $SL(2)$ subalgebra is crucial for the discussion conformal covariance, in particular the appearance of a projective connection. Different $SL(2)$ embeddings lead to various ${\cal W}$-gauge structures. We present a general soldering procedure which allows to express zero curvature conditions for the ${\cal W}$-currents in terms of conformally covariant differential operators acting on the ${\cal W}$ gauge fields and to obtain, at the same time, the complete nilpotent BRS differential algebra generated by ${\cal W}$-currents, gauge fields and the ghost fields corresponding to ${\cal W}$-diffeomorphisms. As illustrations we treat the cases of $SL(2)$ itself and to the two different $SL(2)$ embeddings in $SL(3)$, {\it viz.} the ${\cal W}_3^{(1)}$- and ${\cal W}_3^{(2)}$-gauge structures, in some detail. In these cases we determine algebraically ${\cal W}$-anomalies as solutions of the consistency conditions and discuss their Chern-Simons origin.